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7) which indeed provides one ﬁrst integral [390] K1 = (x2 − 2σ z)e2σ t , b = 2σ . 15)), they all have an even parity1 and it can be proven [283] that they are the only ones in the polynomial class. 19), and an interesting open problem is to ﬁnd a ﬁrst integral, if any, associated to the second solution b = 1 − 3σ of Q2 = 0. In order to ﬁnd ﬁrst integrals of dynamical systems, an original approach has been developed [40], which has no relation with singularities but which takes advantage of basic properties in linear algebra.

105) (n) (0) (n−1) + R (x, u , . . 40), and R(1) is identically zero. 106) without movable critical points. 2). 1). The two main implementations of this perturbation are the Fuchsian perturbative method [149, 82], described hereafter, and the nonFuchsian perturbative method [318], described in the next section. The Fuchsian perturbative method is adapted to the presence of negative integer indices in addition to the ever present value −1, and it generates additional no-log conditions. 108) represents a particular solution containing a number of arbitrary coefﬁcients equal to the number of Fuchs indices, counting their multiplicity.